A Brief Guide to the Great Equations (20 page)

When Maxwell sent the paper to Faraday, Faraday responded that at first he was ‘almost frightened’ by the application of ‘such
mathematical force’ to the subject, but then delighted that the effort succeeded.
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Maxwell’s second step was a paper called ‘On Physical Lines of Force’, written in 1861–62, and it contains one of the greatest uses of analogy in the history of science. Maxwell begins by announcing his intention to ‘examine magnetic phenomena from a mechanical point of view’, and refers to an analogy Thomson had used to understand the Faraday effect: if a magnetic field can shift the plane of polarization of light, Thomson said, it is as if each point on a magnetic line of force were a tiny, spinning ‘molecular vortex’ that passed along some of its spin to any waves of light flowing by.

Maxwell then further develops the image. Let’s say a magnetic field consists of such rotating ‘cells’, as he calls them, whose axes are along magnetic lines of force as if threaded on a string; the stronger the field, the more rapidly the cells spin. But Maxwell knows it is mechanically impossible to have cells on neighboring strings spin the same way – clockwise, let’s say – for those on one string would rub the wrong way against those in the next. Maxwell rescues the picture by assuming that the space in between is filled with something similar to what engineers call ‘idle wheels’ – smaller beads, in contact with the cells, that rotate counterclockwise, permitting the cells to rotate clockwise. These beads stay in place when the neighboring cells are rotating at the same speed, but changes in the speeds of the vortices cause the beads to move in a line, and they are passed from one cell to another. Thus, Maxwell decides, these beads act much like an electric current.

The model displayed the effects of electromagnetism – the way a changing magnetic field generates an electric current, and an electric current generates a magnetic field – as produced by mechanical motions of a medium. Push-pulls in the ether could produce all the electrical and magnetic effects that Faraday and others had noted.
It even produced a mechanical conception of Faraday’s electrotonic state, or what was happening when there was a magnetic field but no electrical current; the electrotonic state was like the impulse of the idle wheels when they turned without moving.

Maxwell’s image

Maxwell’s model

Maxwell wrote up the idea in the spring of 1861, and it was published in installments between March and May. He then left for his usual summer vacation at Glenlair. He was under no illusion that he had created a picture, a representation, of electromagnetism. All he wanted to claim was that this strange model did whatever electrical and magnetic phenomena did, and thus that its mathematics would also work for them. His model was, Maxwell remarked, like an ‘orrery’, or model of the solar system you often see in natural history museums in which the planets are balls placed on rods that mechanically swing about a central ball, the sun. The value of assembling such a model – putting everything you know into it – is that when you finish, and can survey how it works as a whole, you can often see even more than you got from the pieces.

During the vacation, Maxwell realized that he had left something significant out of the model. The cells, he knew, had to have
at least a little springiness or elasticity, as do all solid bodies. But this springiness would cause certain effects in his model that he had not accounted for. When the cells pushed the beads but the beads could not move (in an insulating material, say), the cells’ elasticity would push the beads a little bit anyway, like rubber balls pushing against an immovable force, until the motion is counterbalanced by forces in the material. If the force were removed, the cells and beads would spring back. Maxwell called this a ‘displacement of the electricity’, whose amount depends on the strength of the electromotive force and the nature of the body. He realized he had to incorporate this into his mathematics, which would also involve introducing a small corrective factor to Ampère’s law in the process.

Still more revolutionary: anything elastic can transmit energy from one place to another in the form of waves. Maxwell had shown that the ether – the medium of electrical and magnetic phenomena – must be at least a little elastic. The medium could pass energy in the form of waves from one part to another via leapfrogging electric and magnetic effects operating at right angles to each other – from idle wheels to cells and back to idle wheels again, and on and on, forever. These waves would act the way light does, reflecting, refracting, interfering, and polarizing. Maxwell set out to find the rate that these transverse vibrations travel through the ether, assuming it were passed by purely mechanical forces. The result he calculated, based on the work of Rudolph Kohlrausch and Wilhelm Weber – two German physicists who had measured electrical constants a few years earlier – is 310,740 kilometers, or 193,088 miles, per second. But the velocity of light, as measured by Armand Fizeau a dozen years previously, is 314,858 kilometers, or 195,647 miles, a second, suggestively close. Thus Maxwell wrote, ‘The velocity of transverse undulations in our hypothetical medium, calculated from the electro-magnetic experiments of MM Kohlrausch and Weber, agree so exactly with the velocity of light calculated from the optical experiments of M. Fizeau, that we can scarcely avoid the inference that
light consists in the transverse undulations of the same medium which is
the cause of electric and magnetic phenomena
.’
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He published these two revolutionary new features in the model as Part III of his paper in 1862.

Two years later, Maxwell took a third key step in his paper entitled ‘A Dynamical Theory of the Electromagnetic Field’, written late in 1864 and published early in 1865. In it, he cites the earlier mechanical analogy only to abandon it, aiming to present all the results – including the displacement current and the idea that light is an electromagnetic wave – in the form of a set of freestanding equations. ‘Thus, then, we are led to the conception of a complicated mechanism capable of a vast variety of motion, but at the same time so connected that the motion of one part depends, according to definite relations, on the motion of other parts, these motions being communicated by forces arising from the relative displacement of the connected parts, in virtue of their elasticity. Such a mechanism must be subject to the general laws of Dynamics, and we ought to be able to work out all the consequences of its motion, provided we know the form of the relation between the motions of the parts.’
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Maxwell continued, a few paragraphs later, ‘In order to bring these results within the power of symbolical calculation, I then express them in the form of the General Equations of the Electromagnetic Field.’ He then lists twenty equations in eight general categories.
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This brought to a close one of the most remarkable uses of analogy in science. His achievement is itself often expressed in terms of a famous analogy – ‘Maxwell threw out the bathwater and kept the baby’ – except that the bathwater begat the baby.

In 1873, Maxwell published
A Treatise on Electricity and Magnetism
, his complete presentation of the branch of science that he had developed
by his remarkable analogy, and the form in which practically everyone for at least a decade would have to learn it. About a thousand pages long, it was rather difficult and even annoying to digest, for Maxwell made no effort to condense or simplify the work for the reader, aiming to be comprehensive rather than economical. For instance, in the key chapter, entitled ‘General Equations of the Electromagnetic Field’, Maxwell summarizes his work in twelve steps, labeled A to L, each involving an equation or group of equations. ‘These may be regarded as the principal relations among the quantities we have been considering’, he writes. Some could be combined, ‘but our object is not to obtain compactness in the mathematical formulae.’ Furthermore, these equations were based on concepts that were extremely difficult to use for those interested in practical applications, most notably
A
, the vector potential, and ψ, the scalar potential.

Maxwell’s
Treatise
has also puzzled historians, because in it – and elsewhere – he is silent about how to make and find electromagnetic waves. The idea of electromagnetic waves was the single most thrilling and unexpected feature of Maxwell’s entire life’s work. His silence over how to make and find such waves seems as perverse as an astronomer whose studies predict the existence of a new planet, yet who does not think to go find a telescope to point at it, or tell someone to go do it. Maxwell’s silence is strange enough to demand explanation. Some historians say it is that he was less interested in electromagnetic waves than in light and the ether, others that he did not conceive of any way to produce and detect them, still others that he simply had no time to think on the subject. None of these explanations is really convincing, though it is true that Maxwell’s workload had dramatically increased by the time of the
Treatise
. In 1871, he was given charge of supervising the founding of the new Cavendish Laboratory in Cambridge, England, and in 1874, he was handed the task of editing the papers of the laboratory’s namesake, Henry Cavendish. Maxwell also became the scientific co-editor of the ninth edition of the
Encyclopaedia Britannica
. These projects left him little time for research.

Maxwell did, however, retain his interest in seeing if the ‘great ocean of ether’, as he called it, could somehow be detected. It is invisible and we know little about it. We do not even know, he wrote in his
Encyclopaedia Britannica
entry on ‘Ether’, if dense bodies like the earth pass through this ocean the way fish pass through water, dragging some small portion of it with them; or the ether might pass though them ‘as the water of the sea passes through the meshes of a net when it is towed along by a boat.’ As he wrote beautifully and somewhat anxiously:

There are no landmarks in space; one portion of space is like every other portion, so that we cannot tell where we are. We are, as it were, on an unruffled sea, without stars, compass, soundings, wind, or tide, and we cannot tell in what direction we are going. We have no log which we can cast out to take a dead reckoning by; we may compute our rate of motion with respect to the neighboring bodies, but we do not know how these bodies may be moving in space.
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There is one trick we might play to detect it, he realized, thanks to the fact that a wave flowing through a medium moves at different speeds depending on the speed of the medium. Sound, for instance, always travels at the same speed – about 1,100 feet a second in air – due to the properties of the medium (air molecules) that propagate it. If a wind’s blowing, the sound still travels at the same rate in the air, but because the air carries the sound waves along with it, these will seem to be traveling faster or slower than usual from someone on the ground. If a wind’s blowing, sound waves thus travel at different rates in different directions.

The same should be true of light. In moving around the sun, the earth might ‘drag’ some small amount of ether with it, but would have some changing velocity with respect to the ether; there would be an ether wind or ether drift. The speed of light would be different in different directions. The difference would be tiny – one part in a
hundred million – from the velocity of light in ether at rest. Was this measurable?

On Earth probably not. If experimenters shot beams of light back and forth in different directions, the hundred-millionth-part difference in travel time would be ‘quite insensible’, Maxwell wrote. ‘The only practicable method is to compare the values of the velocity of light deduced from the observation of the eclipses of Jupiter’s satellites when Jupiter is seen from the earth at nearly opposite points of the ecliptic.’ And so in March 1879, he contacted the director of the Nautical Almanac Office, in Cambridge, England, to ask if any research on this subject had been done. ‘I am not an astronomer’, he wrote with his usual modesty in making the inquiry, but ‘the only method, so far as I know’ of measuring the ether drift would be to make precise measurements of the apparent retardation of eclipses of the satellites of Jupiter.
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